Heat Engine and Carnot Cycle Simulator

The area is the answer

Work done by an expanding gas is ∫P dV, the sum of P times each small change in volume, so along any leg of the cycle it is the area under that leg. Travelling to the right the gas expands and does positive work. Coming back to the left it is compressed and the work is negative. What survives the subtraction is precisely the area between the outward and return paths, which is the area enclosed by the loop.

That is why the shaded region is the result rather than a picture of it, and it is also why the direction matters. Traversed clockwise the loop produces net work and the device is an engine. Anticlockwise it consumes work to move heat from cold to hot, and the same diagram describes a refrigerator. No single state variable tells you which you are looking at.

Why there is a ceiling at all

The Carnot efficiency

η = 1 - Tc/Th

depends only on the two temperatures. Not on the gas, not on the machine, not on how cleverly it is built. The argument is short: if some engine beat it, you could use its surplus work to drive a reversed Carnot cycle and end up moving heat from cold to hot with nothing left over, which the second law forbids. The bound is therefore about what is thermodynamically possible rather than about engineering quality.

This has an uncomfortable consequence. To improve efficiency you must widen the temperature gap, and the cold side is usually the surroundings, so in practice you raise the hot side until the materials complain. That, and not inefficiency in the ordinary sense, is why power stations run their steam as hot as they can survive.

Rejected heat is not waste in the ordinary sense

The ledger beside the diagram splits the heat taken in into the part that became work and the part that was dumped. The dumped part shrinks as the cycle gets more efficient, but no arrangement of the cycle removes it.

The reason is that a cycle must return to its starting state to run again. Getting the pressure back down without undoing the work you just extracted means removing energy, and the only place to put it is the cold side. Rejecting heat is the price of repeatability, not a flaw in the design.

Three cycles, and what each one is limited by

Carnot is two isotherms joined by two adiabats, and it sits exactly on the limit. Its problem is practical rather than theoretical: both adiabatic legs span the same temperature range, so both change the volume by the same factor, (Th/Tc)^(1/(γ-1)). At 800 K against 300 K with a diatomic gas that is already 11.6 before the isothermal legs are added. An enormous swept volume for a modest output is most of the reason nobody builds one.

Otto is the petrol engine: two adiabats joined by two constant-volume legs, with combustion supplying the heat at the top of the compression stroke. Its efficiency is 1 - r^(1-γ), which contains no temperature whatsoever. Set the cycle to Otto and sweep the hot side: the efficiency line is flat while the ceiling above it rises. Raising the peak temperature buys nothing; raising the compression ratio buys everything, and knock is what stops you around 11.

Stirling is the interesting one. Two isotherms joined by constant-volume legs, and provided the heat moved on those legs is recovered by a regenerator, it reaches the Carnot efficiency by a completely different route. It is the counterexample to the widespread belief that only a Carnot cycle can. Real regenerators are imperfect, which is where the shortfall in an actual Stirling engine comes from.

Common mistakes

• Thinking efficiency can be improved by using more gas. Every energy scales in proportion, so the ratio does not move. The amount slider demonstrates it.
• Believing only a Carnot cycle can hit the limit. An idealised Stirling cycle does too.
• Reading rejected heat as a defect. It is structural. A cycle that rejected nothing could not return to its starting state.
• Expecting a hotter petrol engine to be more efficient. Otto efficiency has no temperature in it at all.
• Confusing the enclosed area with the area under the curve. Under one leg is that leg’s work. Enclosed by the loop is the net work of the whole cycle.
• Assuming any closed loop is a valid cycle. A loop can close, conserve energy and obey the gas law at every point and still not be the cycle you named. Carnot in particular requires the two isothermal volume ratios to be equal.